Statistics Foundation · Lesson 3.5
Bayes’ theorem and diagnostic reasoning.
Bayes’ theorem continues the conditional probability ideas from Lesson 3.4. Instead of only asking whether information changes a probability, this lesson asks how much the probability should change after evidence is observed. Diagnostic testing provides a powerful example because it forces us to separate prevalence, sensitivity, specificity, false positives and posterior probability.
Lesson route
Move from changed probability to updated probability.
In Lesson 3.4, students asked whether one event changes another event’s probability. In this lesson, students learn the formal updating rule that tells us how to calculate the new probability after evidence arrives.
0–10 min
Continue from conditional probability
Recall from Lesson 3.4 that information can change probability. Bayes’ theorem begins exactly there: after observing evidence, we update the probability of the event we care about.
10–25 min
Identify the reversal problem
Understand why P(A | B) and P(B | A) answer different questions, even though they involve the same two events.
25–45 min
Derive Bayes’ theorem
Use the joint probability identity P(A ∩ B) = P(A | B)P(B) = P(B | A)P(A) to derive the updating formula.
45–70 min
Translate to diagnostic testing
Connect disease prevalence, sensitivity, specificity, false positives and false negatives to the Bayes formula.
70–100 min
Use natural frequencies
Turn percentages into counts so the difference between true positives and false positives becomes visible.
100–130 min
Interpret evidence responsibly
Explain why a positive test can be strong evidence in one population and weak evidence in another population, even when the test is unchanged.
Mastery checklist
Students should be able to update probability with evidence.
Explain why P(A | B) and P(B | A) are different.
Derive Bayes’ theorem from joint probability.
Identify prior, likelihood, evidence and posterior.
Interpret sensitivity and specificity correctly.
Calculate PPV and NPV using natural frequencies.
Explain why prevalence affects diagnostic interpretation.
Recognise the false-positive route in the denominator.
Connect Bayes’ theorem back to dependence from Lesson 3.4.